Session S12: Nonlinear Dynamics: Topology

Symbolic Dynamics Applied to a Chaotic Spherical Vortex

Three-dimensional, time-periodic, volume-preserving flows are common models of complex fluid motion. These flows can exhibit chaotic dynamics with complicated geometric structures. Symbolic dynamics provides an effective tool to describe the underlying topology of these systems. One such approach that has successfully modeled fully 3D volume-preserving flows is Homotopic Lobe Dynamics (HLD). This topological technique was applied to chaotic versions of Hill's spherical vortex. An underlying assumption in past work was that the vortex zone was topologically separated from its exterior. In the present talk, we investigate a series of more challenging examples where this topological condition breaks down. We show that the HLD technique can still describe the topological features of such flows and yield an estimate of the topological entropy.   

Computation of Topological Entropy in Three Dimensions from Fluid Trajectories

We introduce and verify an algorithm for estimating a three-dimensional flow's topological entropy, a measure quantifying the complexity of chaotic dynamics. Analogous to the topological entropy calculation from the "braiding" of system trajectories in two dimensions by Thiffeault, we achieve this in three dimensions by exploiting the collective motion of an ensemble of potentially-sparse system trajectories; as the ensemble evolves in time, the points repeatedly stretch and fold two-dimensional rubber sheets. The topological entropy is bounded below by the exponential growth rate of this sheet, thereby quantifying the flow complexity. New to three-dimensional entropy calculations is the introduction of computational geometry tools: we maintain a Delaunay triangulation of points as they move to record the evolution and growth of a sheet. Because this algorithm requires only trajectory data and no knowledge of governing equations, these results aid greatly in a wide variety of natural and industrial fluid systems, including the large-scale dispersion of pollutants in the Earth's atmosphere and oceans and the rapidly developing field of microfluidics, all while remaining generally applicable to theorists and experimentalists alike.   

Topological Advection

Fluid advection problems typically involve either quantifying mixing (e.g. with topological entropy) or finding barriers to mixing (e.g. by identifying Lagrangian coherent sets). Since topological entropy is given by the exponential stretching rate of material curves and coherent sets are defined to have boundaries which do not appreciably stretch, both advection problems have a common formulation: find the future state of material curves under the action of the flow. When our knowledge of the fluid system is through sparse data (finite set of trajectories), which is the natural output of experiments, this constitutes the problem of topological advection. The current best known topological advection algorithm blends a braid theory representation of trajectories with a clever coordinate system on the space of closed material curves. I will present a new algorithm which solves the topological advection problem more efficiently and perhaps more naturally. It uses ideas from computational geometry to maintain a triangulation of the points as they move, while encoding curves as edge weights that enumerate transverse intersections. These results also naturally extend to higher dimensional versions of this problem.   

Topological Helical Vorticity Compression in Ideal Fluids

We show how an additional topological conserved quantity arises in ideal fluids whenever the helicity vanishes in such a way that the vorticity field is tangent to a family of surfaces. We give examples of vorticity fields for which this quantity does not vanish, and interpret it as measuring helical compression of vortex lines. We show that if this invariant does not vanish then the flow is not steady, giving a topological obstruction for a vorticity field to come from a steady flow. Finally we discuss relations to the Hamiltonian formulation of the Euler equations.   

Topological Flow Data Analysis Part 1- Theory and Applications

We have investigated a mathematical theory classifying the topological structures of streamline patterns for 2D incompressible (Hamiltonian) vector fields on surfaces such as a plane and a spherical surface, in which a unique combinatorial structure, called partially Cyclically Ordered rooted Tree (COT), and its associated graph (Reeb graph) are assigned to every streamline topology. With the COT representations, one can identify the topological streamline structures without ambiguity and predict the possible transition of streamline patterns with a mathematical rigor. In addition, Uda has recently developed a software converting the values of stream function on structured/non-structured grid points in the plane into the COT representation automatically. It enables us to conduct the classification of streamline topologies for a large amount of flow datasets and the snapshots of time-series of flow evolutions obtained by measurements and numerical simulations, which we call Topological Flow Data Analysis (TFDA). In the three consecutive talks Part1-Part3, we introduce the recent developments of TFDA. In the first part followed by two talks, we will present an overview of basic theory and its applications to atmospheric data.   

Topological Flow Data Analysis Part 2- Implementation and Software Demonstration

Topological data analysis attracts researchers in many fields and it also plays important roles in Topological Flow Data Analysis (TFDA). In the TFDA basis by Sakajo and Yokoyama, streamline patterns are identified by partially Cyclically Ordered rooted Trees (COTs) under topological classification. Since their theory is applicable to a wide range of physical phenomena, there is a rise in demand for an algorithm converting given flow data to COTs. A Reeb graph of a real-valued function, which consists of certain topological features, is a central key to the conversion. The author has proposed a new algorithm so that the Reeb graph with certain mathematical properties are obtained. Because our formulation is based on persistent homology in topological data analysis, a stability is ensured in our algorithm. We also establish a consistency theorem that bridges a gap between discrete data and continuous data. Furthermore, we do not assume any interpolation, and hence any mesh-like input data are accepted. Thanks to these nice properties we can effectively construct a COT-representation from discrete flow data. In the second part of our series presentations, we will provide quick explanations for the algorithm and demonstrate usage of a library ``\texttt{psiclone}'' for TFDA.   

Topological Flow Data Analysis Part 3- Foundation of describing flows on 3D spaces

Topologies of flows on 3D spaces are quite complicated. Though every loop (e.g. periodic orbit) separates surfaces but chaotic stationary 2D flows on surfaces do not appear, no loops separate any 3D spaces but chaotic behaviors appear in stationary flows on 3D spaces. Moreover, periodic orbits are unknotted in 2D spaces but they can be easily linked in 3D spaces. Therefore we need the information both of chaos and of linking of infinitely many periodic orbits to analyze flows on 3D spaces. Hence it's very hard to describe topology of flows on 3D spaces in general. However, if there is uniformity (resp. symmetry) of flows, we can use slices of flows on 3D spaces to analyze the topology of such flows as medical doctors use X-ray photographs or computerized tomography images to analyze blood currents. On the other hand, the resulting flows on the slices are not incompressible even if the original 3D flow is incompressible. Thus we need to describe generic 2D flows. Therefore we constructed a classification of generic surface flows, which are called flows of finite type. In the third part of this sequential talks, we will explain the background of representations of 2D flows and introduce a representation of 2D flows of finite type and analyze topologies of flows on 3D spaces.   

Topological equivalence between 3D buoyancy-driven and lid-driven cavity flows

The present study concerns Lagrangian transport and (chaotic) advection in three-dimensional (3D) flows in cavities under steady and laminar conditions. The main goal is to investigate topological equivalences in 3D streamline patterns and their response to nonlinear effects between flow classes driven by different forcing. To this end we consider two classical systems that are important in both natural and industrial applications: a buoyancy-driven flow (laterally-heated configuration) and a lid-driven flow governed by the Grashof (Gr) and the Reynolds (Re) numbers, respectively. Symmetries imply fundamental similarities between the streamline patterns of these flows. Moreover, nonlinearities induced by buoyancy (increasing Gr) in the buoyancy-driven flow versus fluid inertia (increasing Re) and forcing protocol in the lid-driven flow cause similar bifurcations of the flow topology. These analogies imply that Lagrangian transport is governed by universal mechanisms and differences are restricted to the manner in which these phenomena are triggered. Experimental validation of key aspects of the Lagrangian dynamics is carried out by particle image velocimetry and 3D particle-tracking velocimetry.